If those are Chinese tires it would not surprise me at all. 

Likely made in China or Mexico and wouldn't be out of the question that they can't be balanced. 

How long have you had the tires?

How many miles are on them? 

Have they developed a bad wear pattern due to being driven while unbalanced or misaligned?

You state that they may be defective. Are they still under warranty?

Has the tire shop tried “road force balancing”?

Go to a different shop and see if they can.

If they are unable to balance them either - ask them to try and balance the rims without the tires.

What you're describing sounds like either a scam to sell you tires, broken equipment or very low quality personal.

They did the best they could but said they could not balance them all the way

Is it possible for tires to not be possible to balance all the way?

There's a select numbers way in which you can fail to perform a balance. In what way did they fail?

Was the required weight to add too high? Or did it constantly shift when they attempted adding it?

It's hard to make a determination without that info.

Unbalanceable tires? no, super unlikely - assuming they're in good shape.

Wheels? if they're severely damaged, but then it would be visible.

My personal suspicion is that the guys in that tire shops might be clowns.